Theorem xlt2add 10114

Description: Extended real version of lt2add 8624. Note that ltleadd 8625, which has weaker assumptions, is not true for the extended reals (since 0 + +oo < 1 + +oo fails). (Contributed by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
xlt2add	|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A < C /\ B < D ) -> ( A +e B ) < ( C +e D ) ) )

Proof of Theorem xlt2add

Step	Hyp	Ref	Expression
1		xaddcl 10094	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) e. RR* )
2	1	3ad2ant1 1044	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) e. RR* )
3	2	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e B ) e. RR* )
4		simp1l 1047	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A e. RR* )
5		simp2r 1050	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> D e. RR* )
6		xaddcl 10094	. . . . . . . 8 |- ( ( A e. RR* /\ D e. RR* ) -> ( A +e D ) e. RR* )
7	4, 5, 6	syl2anc 411	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e D ) e. RR* )
8	7	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e D ) e. RR* )
9		xaddcl 10094	. . . . . . . 8 |- ( ( C e. RR* /\ D e. RR* ) -> ( C +e D ) e. RR* )
10	9	3ad2ant2 1045	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( C +e D ) e. RR* )
11	10	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( C +e D ) e. RR* )
12		simp3r 1052	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B < D )
13	12	adantr 276	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> B < D )
14		simp1r 1048	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B e. RR* )
15	14	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> B e. RR* )
16	5	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> D e. RR* )
17		simprl 531	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> A e. RR )
18		xltadd2 10111	. . . . . . . 8 |- ( ( B e. RR* /\ D e. RR* /\ A e. RR ) -> ( B < D <-> ( A +e B ) < ( A +e D ) ) )
19	15, 16, 17, 18	syl3anc 1273	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( B < D <-> ( A +e B ) < ( A +e D ) ) )
20	13, 19	mpbid 147	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e B ) < ( A +e D ) )
21		simp3l 1051	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A < C )
22	21	adantr 276	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> A < C )
23	4	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> A e. RR* )
24		simp2l 1049	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> C e. RR* )
25	24	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> C e. RR* )
26		simprr 533	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> D e. RR )
27		xltadd1 10110	. . . . . . . 8 |- ( ( A e. RR* /\ C e. RR* /\ D e. RR ) -> ( A < C <-> ( A +e D ) < ( C +e D ) ) )
28	23, 25, 26, 27	syl3anc 1273	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A < C <-> ( A +e D ) < ( C +e D ) ) )
29	22, 28	mpbid 147	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e D ) < ( C +e D ) )
30	3, 8, 11, 20, 29	xrlttrd 10043	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e B ) < ( C +e D ) )
31	30	anassrs 400	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) /\ D e. RR ) -> ( A +e B ) < ( C +e D ) )
32		pnfxr 8231	. . . . . . . . . . . 12 |- +oo e. RR*
33	32	a1i 9	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> +oo e. RR* )
34		pnfge 10023	. . . . . . . . . . . 12 |- ( C e. RR* -> C <_ +oo )
35	24, 34	syl 14	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> C <_ +oo )
36	4, 24, 33, 21, 35	xrltletrd 10045	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A < +oo )
37		npnflt 10049	. . . . . . . . . . 11 |- ( A e. RR* -> ( A < +oo <-> A =/= +oo ) )
38	4, 37	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A < +oo <-> A =/= +oo ) )
39	36, 38	mpbid 147	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A =/= +oo )
40		pnfge 10023	. . . . . . . . . . . 12 |- ( D e. RR* -> D <_ +oo )
41	5, 40	syl 14	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> D <_ +oo )
42	14, 5, 33, 12, 41	xrltletrd 10045	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B < +oo )
43		npnflt 10049	. . . . . . . . . . 11 |- ( B e. RR* -> ( B < +oo <-> B =/= +oo ) )
44	14, 43	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( B < +oo <-> B =/= +oo ) )
45	42, 44	mpbid 147	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B =/= +oo )
46		xaddnepnf 10092	. . . . . . . . 9 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
47	4, 39, 14, 45, 46	syl22anc 1274	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) =/= +oo )
48		npnflt 10049	. . . . . . . . 9 |- ( ( A +e B ) e. RR* -> ( ( A +e B ) < +oo <-> ( A +e B ) =/= +oo ) )
49	2, 48	syl 14	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( ( A +e B ) < +oo <-> ( A +e B ) =/= +oo ) )
50	47, 49	mpbird 167	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) < +oo )
51	50	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = +oo ) -> ( A +e B ) < +oo )
52		oveq2 6025	. . . . . . 7 |- ( D = +oo -> ( C +e D ) = ( C +e +oo ) )
53		mnfxr 8235	. . . . . . . . . . 11 |- -oo e. RR*
54	53	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo e. RR* )
55		mnfle 10026	. . . . . . . . . . 11 |- ( A e. RR* -> -oo <_ A )
56	4, 55	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo <_ A )
57	54, 4, 24, 56, 21	xrlelttrd 10044	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo < C )
58		nmnfgt 10052	. . . . . . . . . 10 |- ( C e. RR* -> ( -oo < C <-> C =/= -oo ) )
59	24, 58	syl 14	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo < C <-> C =/= -oo ) )
60	57, 59	mpbid 147	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> C =/= -oo )
61		xaddpnf1 10080	. . . . . . . 8 |- ( ( C e. RR* /\ C =/= -oo ) -> ( C +e +oo ) = +oo )
62	24, 60, 61	syl2anc 411	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( C +e +oo ) = +oo )
63	52, 62	sylan9eqr 2286	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = +oo ) -> ( C +e D ) = +oo )
64	51, 63	breqtrrd 4116	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = +oo ) -> ( A +e B ) < ( C +e D ) )
65	64	adantlr 477	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) /\ D = +oo ) -> ( A +e B ) < ( C +e D ) )
66		simpr 110	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = -oo ) -> D = -oo )
67		mnfle 10026	. . . . . . . . . 10 |- ( B e. RR* -> -oo <_ B )
68	14, 67	syl 14	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo <_ B )
69	54, 14, 5, 68, 12	xrlelttrd 10044	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo < D )
70		nmnfgt 10052	. . . . . . . . 9 |- ( D e. RR* -> ( -oo < D <-> D =/= -oo ) )
71	5, 70	syl 14	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo < D <-> D =/= -oo ) )
72	69, 71	mpbid 147	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> D =/= -oo )
73	72	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = -oo ) -> D =/= -oo )
74	66, 73	pm2.21ddne 2485	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = -oo ) -> ( A +e B ) < ( C +e D ) )
75	74	adantlr 477	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) /\ D = -oo ) -> ( A +e B ) < ( C +e D ) )
76		elxr 10010	. . . . . 6 |- ( D e. RR* <-> ( D e. RR \/ D = +oo \/ D = -oo ) )
77	5, 76	sylib 122	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( D e. RR \/ D = +oo \/ D = -oo ) )
78	77	adantr 276	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) -> ( D e. RR \/ D = +oo \/ D = -oo ) )
79	31, 65, 75, 78	mpjao3dan 1343	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) -> ( A +e B ) < ( C +e D ) )
80		simpr 110	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = +oo ) -> A = +oo )
81	39	adantr 276	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = +oo ) -> A =/= +oo )
82	80, 81	pm2.21ddne 2485	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = +oo ) -> ( A +e B ) < ( C +e D ) )
83		oveq1 6024	. . . . 5 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
84		xaddmnf2 10083	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
85	14, 45, 84	syl2anc 411	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo +e B ) = -oo )
86	83, 85	sylan9eqr 2286	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = -oo ) -> ( A +e B ) = -oo )
87		xaddnemnf 10091	. . . . . . 7 |- ( ( ( C e. RR* /\ C =/= -oo ) /\ ( D e. RR* /\ D =/= -oo ) ) -> ( C +e D ) =/= -oo )
88	24, 60, 5, 72, 87	syl22anc 1274	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( C +e D ) =/= -oo )
89		nmnfgt 10052	. . . . . . 7 |- ( ( C +e D ) e. RR* -> ( -oo < ( C +e D ) <-> ( C +e D ) =/= -oo ) )
90	10, 89	syl 14	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo < ( C +e D ) <-> ( C +e D ) =/= -oo ) )
91	88, 90	mpbird 167	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo < ( C +e D ) )
92	91	adantr 276	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = -oo ) -> -oo < ( C +e D ) )
93	86, 92	eqbrtrd 4110	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = -oo ) -> ( A +e B ) < ( C +e D ) )
94		elxr 10010	. . . 4 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
95	4, 94	sylib 122	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
96	79, 82, 93, 95	mpjao3dan 1343	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) < ( C +e D ) )
97	96	3expia 1231	1 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A < C /\ B < D ) -> ( A +e B ) < ( C +e D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1003   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   class class class wbr 4088  (class class class)co 6017   RRcr 8030   +oocpnf 8210   -oocmnf 8211   RR*cxr 8212   < clt 8213   <_ cle 8214   +ecxad 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-xadd 10007

This theorem is referenced by:  bldisj  15124